HES 505 Fall 2022: Session 19
Matt Williamson
By the end of today you should be able to:
Define the weights of evidence approach to overlays
Recognize the link between regression analysis and overlay analysis
Generate spatial predictions based on regression analysis
Extend logistic regression to presence-only data models
\[ \begin{equation} F(\mathbf{s}) = \prod_{M=1}^{m}X_m(\mathbf{s}) \end{equation} \]
\[ \begin{equation} F(\mathbf{s}) = f(w_1X_1(\mathbf{s}), w_2X_2(\mathbf{s}), w_3X_3(\mathbf{s}), ..., w_mX_m(\mathbf{s})) \end{equation} \]
\(F(\mathbf{s})\) does not have to be binary (could be ordinal or continuous)
\(X_m(\mathbf{s})\) could also be extended beyond simply ‘suitable/not suitable’
Adding weights allows incorporation of relative importance
Other functions for combining inputs (\(X_m(\mathbf{s})\))
\[ \begin{equation} F(\mathbf{s}) = \frac{\sum_{i=1}^{m}w_iX_i(\mathbf{s})}{\sum_{i=1}^{m}w_i} \end{equation} \]
\(F(s)\) is now an index based of the values of \(X_m(\mathbf{s})\)
\(w_i\) can incorporate weights of evidence, uncertainty, or different participant preferences
Dividing by \(\sum_{i=1}^{m}w_i\) normalizes by the sum of weights
\[ \begin{equation} F(\mathbf{s}) = w_0 + \sum_{i=1}^{m}w_iX_i(\mathbf{s}) + \epsilon \end{equation} \]
If we estimate \(w_i\) using data, we specify \(F(s)\) as the outcome of regression
When \(F(s)\) is binary → logistic regression
When \(F(s)\) is continuous → linear (gamma) regression
When \(F(s)\) is discrete → Poisson regression
Assumptions about \(\epsilon\) matter!!